We are going to analyze the interval solution of an elastic beam under uncertain boundary conditions. Boundary conditions are defined as rotational springs presenting interval stiffness. Developments occur according to the interval analysis theory, which is affected, at the same time, by the overestimation of interval limits (also known as overbounding, because of the propagation of the uncertainty in the model). We suggest a method which aims to reduce such an overestimation in the uncertain solution. This method consists in a reparameterization of the closed form Euler-Bernoulli solution and set intersection.

While dealing with real problem solutions, during some structural engineering procedures, uncertainties must be taken into consideration.

One example is the presence of epistemic uncertainties due to lack of knowledge for what concerns physical phenomena under study. This is an issue which could be included among the problem equations under the label of uncertain parameters.

At this stage it is necessary to introduce a representation of such parameters according to a theory of uncertainty.

The classical way of proceeding is to represent aleatory uncertainties as stochastic variables and solving the problem in two different ways: through a probabilistic analysis if equations are allowed to be integrated or through a Monte Carlo approach, if a numerical solution is required.

Nowadays the increasing tendency is to represent epistemic uncertainty through means of nonprobabilistic methods with an uncertain interval analysis [

In particular interval analysis [

It is commonly known that the main drawback of interval analysis is the so-called dependency effect, which leads to a deep overestimation (also called overbounding) of the interval solution bounds.

This could represent a serious problem since, even though all possible solutions are included with certainty, very large bounds generally make the result meaningless from a physical point of view, and hence useless from an engineering point of view. This is the reason why there is a diffused attempt, together with the community of people who deal with interval analysis based applications, to reduce such overbounding [

In this paper we propose a method in order to reduce the interval overestimation, based on a varied parameterization of functions’ domain (that is reparameterization) and interval set intersections. Our discussion about this method is based on its application to a classical benchmark, that is, the simple solution of an Euler-Bernoulli elastic beam with uncertain boundary conditions, by providing theoretical developments and numerical example. The proposed procedure is compared with a standard interval solution and well-known interval approach, called interval hull method which has been used to decrease the overestimation. Computational issues will be discussed, as well.

One key point is the definition of interval functions, called interval extensions. As Moore affirms in his book [

In the following we denote, in bold letters, any uncertain quantity as an interval

The main purpose of the evaluation of interval functions is to estimate the function range

Considering a real valued function

Interval extensions are not unique. For example, for every

In general differences in the functional form of natural extensions lead to differences in the uncertainty propagation, that finally affect the uncertainty associated with the estimation of the function range

For example we can consider the functions representation in Figure

(a) Different overbounding of interval function. (b) Overbounding reduction by interval hull method (HM).

On the other hand, Figure

From now on the hull method will be used as the main solution for any other confrontation with the proposed method.

In this section, we discuss the use of interval analysis in solving the simple structural problem of an elastic beam with uncertain boundary conditions, represented by elastic springs endowed with interval stiffness. In many applications structural joints are very different with respect to the way that they are modeled (e.g., elastic springs) and joints parameter identification is a key problem in the modeling of complex structures [

The uncertain quantities are inserted in the problem equations as interval variables and the provided solution is discussed according to the theory of interval analysis [

The problem equations follow from the static solution of a supported beam with rotational springs at ends (Figure

The selected structural problem.

The uncertain spring parameters

Here we present the standard interval solution that is found by using the interval natural extension of the closed form solution of the problem in Figure

Figure ^{2},

Interval solution of the displacement field

It is evident in Figure

The correction of the solution given in the previous section is here proposed through a reparameterization of (

From this position (

From the interval analysis point of view, interval functions defined by (

Due to the nonuniqueness of the interval extension, different choices of

Different displacement fields obtained for

It can be noticed, from Figure

Denoting as

Equation (

Figure

RVI solution versus HM solution (

From Figure

A new interval method aimed at reducing interval overbounding has been presented and exemplified through the application to a specific class of structural problems. In particular it is applied to find the sharp uncertain displacement field of beams with uncertain boundary conditions. The method (RPI) is based on the domain reparameterization of the analytical expression of the solution and is able to significantly reduce the overbounding by preserving some physical assumption such as fixity in different boundary conditions. The effectiveness of the presented method is showed by comparison with a classical literature method (HM).

The method is here presented for the cases when a closed form solution of the problem is known. This limitation does not put in jeopardy the qualitative results, which could be further generalized for the application to finite element methods, acting on the element shape functions whose expressions are known. Further developments will be hence devoted to generalize the Interval Finite Element method for beam structures and will allow for applications to every kind of framed structures, where boundary conditions (external or between elements) are uncertain.

The authors declare that there is no conflict of interests regarding the publication of this paper.